What is the largest prime factor of $1337$?
Solution: $1337$ happens to be divisible by $7$. A fairly fast way to see this might be to note that $140-7=133$ is divisible by $7$, so $1330$ must be as well.  Therefore, so is $1330+7= 1337$. Divide $1337$ by $7$ to get 191.  Since 191 is not divisible by 2, 3, 5, 7, 11, 13, or 17, and $17^2 = 289$ is greater than 191, we know that $191$ is prime.  So, the prime factorization of 1337 is $7\cdot 191$, which means the largest prime factor of 1337 is $\boxed{191}$.